On a class of finite solvable groups

James Beidleman; Hermann Heineken; Jack Schmidt

Open Mathematics (2013)

  • Volume: 11, Issue: 9, page 1598-1604
  • ISSN: 2391-5455

Abstract

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A finite solvable group G is called an X-group if the subnormal subgroups of G permute with all the system normalizers of G. It is our purpose here to determine some of the properties of X-groups. Subgroups and quotient groups of X-groups are X-groups. Let M and N be normal subgroups of a group G of relatively prime order. If G/M and G/N are X-groups, then G is also an X-group. Let the nilpotent residual L of G be abelian. Then G is an X-group if and only if G acts by conjugation on L as a group of power automorphisms.

How to cite

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James Beidleman, Hermann Heineken, and Jack Schmidt. "On a class of finite solvable groups." Open Mathematics 11.9 (2013): 1598-1604. <http://eudml.org/doc/269133>.

@article{JamesBeidleman2013,
abstract = {A finite solvable group G is called an X-group if the subnormal subgroups of G permute with all the system normalizers of G. It is our purpose here to determine some of the properties of X-groups. Subgroups and quotient groups of X-groups are X-groups. Let M and N be normal subgroups of a group G of relatively prime order. If G/M and G/N are X-groups, then G is also an X-group. Let the nilpotent residual L of G be abelian. Then G is an X-group if and only if G acts by conjugation on L as a group of power automorphisms.},
author = {James Beidleman, Hermann Heineken, Jack Schmidt},
journal = {Open Mathematics},
keywords = {S-permutable subgroup; PST-group; Normalizer; X-group; permutable subgroups; system normalizers; Sylow subgroups; nilpotent residuals; subgroup closed classes of groups; factor closed classes of groups},
language = {eng},
number = {9},
pages = {1598-1604},
title = {On a class of finite solvable groups},
url = {http://eudml.org/doc/269133},
volume = {11},
year = {2013},
}

TY - JOUR
AU - James Beidleman
AU - Hermann Heineken
AU - Jack Schmidt
TI - On a class of finite solvable groups
JO - Open Mathematics
PY - 2013
VL - 11
IS - 9
SP - 1598
EP - 1604
AB - A finite solvable group G is called an X-group if the subnormal subgroups of G permute with all the system normalizers of G. It is our purpose here to determine some of the properties of X-groups. Subgroups and quotient groups of X-groups are X-groups. Let M and N be normal subgroups of a group G of relatively prime order. If G/M and G/N are X-groups, then G is also an X-group. Let the nilpotent residual L of G be abelian. Then G is an X-group if and only if G acts by conjugation on L as a group of power automorphisms.
LA - eng
KW - S-permutable subgroup; PST-group; Normalizer; X-group; permutable subgroups; system normalizers; Sylow subgroups; nilpotent residuals; subgroup closed classes of groups; factor closed classes of groups
UR - http://eudml.org/doc/269133
ER -

References

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